A conservative Allen-Cahn equation with a space-time dependent Lagrange multiplier

Junseok Kim, Seunggyu Lee, Yongho Choi

Research output: Contribution to journalArticlepeer-review

96 Citations (Scopus)


We present a new numerical scheme for solving a conservative Allen-Cahn equation with a space-time dependent Lagrange multiplier. Since the well-known classical Allen-Cahn equation does not have mass conservation property, Rubinstein and Sternberg introduced a nonlocal Allen-Cahn equation with a time dependent Lagrange multiplier to enforce conservation of mass. However, with their model it is difficult to keep small features since they dissolve into the bulk region. One of the reasons for this is that mass conservation is realized by a global correction using the time-dependent Lagrange multiplier. To resolve the problem, we use a space-time dependent Lagrange multiplier to preserve the volume of the system and propose a practically unconditionally stable hybrid scheme to solve the model. The numerical results indicate a potential usefulness of our proposed numerical scheme for accurately calculating geometric features of interfaces.

Original languageEnglish
Pages (from-to)11-17
Number of pages7
JournalInternational Journal of Engineering Science
Publication statusPublished - 2014 Nov

Bibliographical note

Funding Information:
The first author (J.S. Kim) was supported by a Korea University Grant. The corresponding author (Y. Choi) thanks Dr. D. Jeong and D. Lee for many useful discussions. The authors also wish to thank the reviewers for the constructive and helpful comments on the revision of this article.


  • Allen-Cahn equation
  • Finite difference method
  • Mass conservation
  • Multigrid method
  • Operator splitting

ASJC Scopus subject areas

  • General Materials Science
  • General Engineering
  • Mechanics of Materials
  • Mechanical Engineering


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